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The SIunits package
support for the International System of Units
Marcel Heldoorn†
SIunits@webschool.nl
File date 2002/08/01 — Printed August 1, 2002
Abstract This article describes the SIunits package that provides support for the Systeme International d’Unit´es (SI). The Systeme International d’Unit´es (SI), the modern form of the metric system, is the most widely used system of units and measures around the world. But despite this there is widespread misuse of the system with incor- rect names and symbols used as a matter a course - even by well educated and trained people who should know better. For example how often do we see: mHz, Mhz or mhz written when referring to computer clock rates? The correct form is actually MHz. Note that the capitalisation does matter. Hence, a clear system for the use of units is needed, satisfying the next principles:
- the system should consist of measuring units based on unvariable quan- tities in nature;
- all units other than the base units should be derived from these base units; and
- multiples and submultiples of the units should be decimal. The name Syst`eme International d’Unit´es (International System of Units) with the international abbreviation SI was adopted by the Conf´erence G´en´erale des Poids et Mesures (CGPM) in 1960. It is a coherent system based on seven base units (CGPM 1960 and 1971). The SIunits package can be used to standardise the use of units in your writings. Most macros are easily adaptable to personal preferences. Howe- ver, you are welcome (and strongly invited^1 ) to suggest any improvements.
Enjoy the SIunits package! marcel h.
∗This file has version number v1.33, last revised 2002/08/ †Mail: Kennedylaan 24, NL-3844 BC Harderwijk, The Netherlands (^1) There is an enormous LATEX Knowledge Base out there.
What’s new?
New in version v1.
- Adaptive spacing of \degree, \paminute, \arcminute, \pasecond and \arcsecond: no space between quantity and unit.
- Solved bug: extra space in ready to use ready-to-use units when using texts- tyle option. Thanks to Svend Tollak Munkejord.
- Option italian added to solve interference problem with the babel package and language italian: Babel defines \unit. When using the option italian, use \unita instead. Thanks to Lorenzo Cappelletti and Luca Rossato.
New in version 1.x
- binary.sty style with binary prefixes and units added (see table 6);
- binary.sty can be loaded by using the binary package option (see page 24);
- unit \one added: the derived unit for a derived quantity of dimension one is also the number one;
- In the pstricks package the command \gray is defined. This will cause error messages when the pstricks package is used in combination with the SIunits package. To prevent errors one can choose two different options:
pstricks This option redefines the pstricks command \gray to get the desired SIunits definition of the command. Note: When using this option, the pstricks command \gray is redefined. Gray This option defines a new command \Gray that can be used in- stead of the SIunits command \gray. Note: When using this option, \gray is defined in the pstricks package.
- When using the option textstyle units are printed in the typeface of the enclosing text, automatically.
- the. (period) was made active in the second argument of the \unit macro: it will act like a unit skip (\usk), for example: use \unit{1}{\newton.\metre} instead of \unit{1}{\newton\usk\metre}.
- \katal added: “The 21st Conf´erence G´en´erale des Poids et Mesures deci- des to adopt the special name katal, symbol kat, for the SI unit mole per second to express catalytic activity, especially in the fields of medicine and biochemistry, ...” (21th CGPM (1999), Resolution 12).
- The ready-to-use units used \square instead of \squaren when using the option squaren. Fixed!
- Fixed index and change history generation error.
- Documentation update: implementation of SI-brochure Supplement 2000.
Contents
1 Introduction
1.1 Historical notes
In 1948 the 9th General Conference on Weights and Measures (CGPM^2 ), by its Resolution 6, instructed the International Committee for Weights and Measures (CIPM^2 ):
‘to study the establishment of a complete set of rules for units of me- asurement’; ‘to find out for this purpose, by official inquiry, the opinion prevailing in scientific, technical, and educational circles in all countries’; and ‘to make recommendations on the establishment of a practical system of units of measurement suitable for adoption by all signatories to the Meter Convention.’
The same General Conference also laid down, by its Resolution 7, general principles for unit symbols and also gave a list of units with special names. The 10th CGPM (1954), by its Resolution 6, and the 14th CGPM (1971), by its Resolution 3, adopted as base units of this ‘practical system of units,’ the units of the following seven quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. The 11th CGPM (1960), by its Resolution 12, adopted the name Syst`eme International d’Unit´es (International System of Units), with the international ab- breviation SI, for this practical system of units of measurement, and laid down rules for the prefixes, the derived and supplementary units, and other matters, thus establishing a comprehensive specification for units of measurement.
1.2 The classes of SI units
The General Conference decided to base the International System on a choice of seven well-defined units which by convention are regarded as dimensionally independent: the metre, the kilogram, the second, the ampere, the kelvin, the mole, and the candela. These units are called base units. The second class of SI units contain derived units, i. e., units that can be formed by combining base units according to the algebraic relations linking the corresponding quantities. The names and symbols of some units thus formed in terms of base units can be replaced by special names and symbols which can themselves be used to form expressions and symbols of other derived units (see section 2.2, p. 12). The 11th CGPM (1960) admitted a third class of SI units, called supplementary units and containing the SI units of plane and solid angle. The 20th CGPM (1995) decided to eliminate the class of supplementary units as a separate class in the SI. Thus the SI now consists of only two classes of units: base units and derived units, with the radian and the steradian, which are the two supplementary units, subsumed into the class of derived SI units.
(^2) See section 1.4 for acronyms
1.3 The SI prefixes
The General Conference has adopted a series of prefixes to be used in forming the decimal multiples and submultiples of SI units. Following CIPM Recommendati- on 1 (1969), the set of prefixes is designated by the name SI prefixes. The multiples and submultiples of SI units, which are formed by using the SI prefixes, should be designated by their complete name, multiples and submultiples of SI units, in order to make a distinction between them and the coherent set of SI units proper.
1.4 Acronyms
The SI was established in 1960 by the CGPM. The CGPM is an intergovernmental treaty organisation created by a diplomatic treaty called the Meter Convention (Convention du Metre, often called the Treaty of the Meter in the United States). The Meter Convention was signed in Paris in 1875 by representatives of seventeen nations, including the United States. There are now forty-eight Member States, including all the major industrialised countries. The Convention, modified slightly in 1921, remains the basis of all international agreement on units of measurement. The Meter Convention also created the International Bureau of Weights and Measures (BIPM, Bureau International des Poids et Mesures) and the Internatio- nal Committee for Weights and Measures (CIPM, Comit´e International des Poids et Mesures). The BIPM, which is located in Sevres, a suburb of Paris, France, and which has the task of ensuring worldwide unification of physical measurements, operates under the exclusive supervision of the CIPM, which itself comes under the authority of the CGPM.
CGPM General Conference on Weights and Measures (Conf´erence G´en´erale des Poids et Mesures). The CGPM is the primary intergovernmental treaty or- ganisation responsible for the SI, representing nearly 50 countries. It has the responsibility of ensuring that the SI is widely disseminated and modifying it as necessary so that it reflects the latest advances in science and technology.
CIPM International Committee for Weights and Measures (Comit´e Internatio- nal des Poids et Mesures). The CIPM comes under the authority of the CGPM. It suggests modifications to the SI to the CGPM for formal adopti- on. The CIPM may also on its own authority pass clarifying resolutions and recommendations regarding the SI.
BIPM International Bureau of Weights and Measures (Bureau International des Poids et Mesures). The BIPM, located outside Paris, has the task of en- suring worldwide unification of physical measurements. It is the “interna- tional” metrology institute, and operates under the exclusive supervision of the CIPM.
1.5 Some useful definitions
quantity in the general sense A quantity in the general sense is a property ascribed to phenomena, bodies, or substances that can be quantified for, or assigned to, a particular phenomenon, body, or substance. Examples are mass and electric charge.
kilogram; kilogramme
Le kilogramme est l’unit´e de masse; il est ´egal `a la masse du prototype interna- tional du kilogramme. (1st CGPM (1889) and 3rd CGPM (1901)).
The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. Note: This international prototype is made of platinum-iridium and is kept at the International Bureau of Weights and Measures, S`evres, France.
second; seconde
La seconde est la dur´ee de 9 192 631 770 p´eriodes de la radiation correspondant `a la transition entre les deux niveaux hyperfins de l’´etat fondamental de l’atome de cesium 133. (13th CGPM (1967), Resolution 1).
The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom.
Note: This definition refers to a caesium atom at rest at a temperature of 0 K.
ampere; amp`ere
L’ampere est l’intensit´e d’un courant constant qui, maintenu dans deux con- ducteurs paralleles, rectilignes, de longueur infinie, de section circulaire n´egligeable, et plac´es a une distance de 1 metre l’un de l’autre dans le vide, produirait entre ces conducteurs une force ´egale a 2 × 10 −^7 newton par metre de longueur. (9th CGPM (1948), Resolutions 2 and 7).
The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10 −^7 newton per metre of length.
kelvin; kelvin
Le kelvin, unit´e de temp´erature thermodynamique, est la fraction 1 / 273. 16 de la temp´erature thermodynamique du point triple de l’eau. (13th CGPM (1967), Resolution 4).
The kelvin, unit of thermodynamic temperature, is the fraction 1/ 273 .16 of the thermodynamic temperature of the triple point of water.
Note: The 13th CGPM (1967, Resolution 3) also decided that the unit kelvin and its symbol K should be used to express both thermodynamic temperature and an interval or a difference of temperature, instead of ‘degree Kelvin’ with symbol ◦K. In addition to the thermodynamic temperature (symbol T ) there is also the Celsius (symbol t) defined by the equation t = T − T 0 where T 0 = 273.15 K. Celsius temperature is expressed in degree Celsius; degr´e Celsius (symbol ◦C).
The unit ‘degree Celsius’ is equal to the unit ‘kelvin’; in this case, ‘degree Celsius’ is a special name used in place of ‘kelvin’. A temperature interval or difference of Celsius temperature can, however, be expressed in kelvins as well as in degrees Celsius.
mole; mole
- La mole est la quantit´e de mati
ere d’un systeme contenant autant d’entit´es ´el´ementaires qu’il y a d’atomes dans 0 , 012 kilogramme de carbone 12. - Lorsqu’on emploie la mole, les entit´es ´el´ementaires doivent ˆetre sp´ecifi´ees et peuvent ˆetre des atomes, des mol´ecules, des ions, des ´electrons, d’autres par- ticules ou des groupements sp´ecifi´es de telles particules. (14th CGPM (1971), Resolution 3).
- The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.
- When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles or specified groups of such particle.
Note: In this definition, it is understood that the carbon 12 atoms are unbound, at rest and in their ground state.
candela; candela
La candela est l’intensit´e lumineuse, dans une direction donn´ee, d’une source qui ´emet une radiation monochromatique de fr´equence 540 × 1012 hertz et dont l’intensit´e ´energ´etique dans cette direction est 1 / 683 watt par st´eradian. (16th CGPM (1979), Resolution 3).
The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of a frequency 540 × 1012 hertz and has a radiant intensity in that direction of 1/683 watt per steradian.
2.2.2 SI derived units with special names and symbols
For ease of understanding and convenience, 21 SI derived units have been given special names and symbols, as shown in table 3. They may themselves be used to express other derived units.
2.2.3 Use of SI derived units with special names and symbols
Examples of SI derived units that can be expressed with the aid of SI derived units having special names and symbols (including the radian and steradian) are given in table 3. The advantages of using the special names and symbols of SI derived units are apparent in table 4. Consider, for example, the quantity molar entropy: the unit J/mol K is obviously more easily understood than its SI base-unit equivalent, m^2 kg s−^2 K−^1 mol−^1. Nevertheless, it should always be recognised that the special names and symbols exist for convenience. Tables 3 & 4 also show that the values of several different quantities are expressed in the same SI unit. For example, the joule per kelvin (J/K) is the SI unit for heat capacity as well as for entropy. Thus the name of the unit is not sufficient to define the quantity measured. A derived unit can often be expressed in several different ways through the use of base units and derived units with special names. In practice, with certain quantities, preference is given to using certain units with special names, or combinations of units, to facilitate the distinction between quantities whose values have identical expressions in terms of SI base units. For example, the SI unit of frequency is specified as the hertz (Hz) rather than the reciprocal second (s−^1 ), and the SI unit of moment of force is specified as the newton metre (N m) rather than the joule (J).
2.3 Dimension of a quantity
Any SI derived quantity Q can be expressed in terms of the SI base quantities length (l), mass (m), time (t), electric current (I), thermodynamic temperature (T ), amount of substance (n), and luminous intensity (Iv) by an equation of the form
Q = lαmβ^ tγ^ Iδ^ T εnζ^ I vη
∑^ K
k=
ak,
where the exponents α, β, γ,... are numbers and the factors ak are also numbers. The dimension of Q is defined to be
dim Q = LαMβ^ Tγ^ Iδ^ ΘεNζ^ Jη^ ,
where L, M, T, I, Θ, N and J are the dimensions of the SI base quantities length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity, respectively. The exponents α, β, γ,... are called “dimen- sional exponents”. The SI derived unit of Q is mα^ kgβ^ sγ^ Aδ^ Kε^ molζ^ cdη^ , which is obtained by replacing the dimensions of the SI base quantities in the dimension of Q with the symbols for the corresponding base units.
For example: Consider a nonrelativistic particle of mass m in uniform motion which travels a distance l in a time t. Its velocity is υ = l/t and its kinetic energy is Ek = mυ^2 /2 = l^2 mt−^2 /2. The dimension of Ek is dim Ek = L^2 MT−^2 and the dimensional exponents are 2, 1, and −2.
Table 3: — SI derived units with special names and symbols — Name Expression in Symbol Expression in SI base units SI derived units radiana^ m m−^1 = 1b^ rad m m−^1 steradiana^ m^2 m−^2 = 1b^ src^ m^2 m−^2 hertz s−^1 Hz s−^1 newton m kg s−^2 N m kg s−^2 pascal m−^1 kg s−^2 Pa N m−^2 joule m^2 kg s−^2 J N m watt m^2 kg s−^3 W J s−^1 coulomb A s C A s volt m^2 kg s−^3 A−^1 V W A−^1 farad m−^2 kg−^1 s^4 A^2 F C V−^1 ohm m^2 kg s−^3 A−^2 Ω V A−^1 siemens m−^2 kg−^1 s^3 A^2 S A V−^1 weber m^2 kg s−^2 A−^1 Wb m^2 kg s−^2 A−^1 tesla kg s−^2 A−^1 T Wb m−^2 henry m^2 kg s−^2 A−^2 H Wb A−^1 celsius K ◦C K lumen cd m^2 m−^2 c^ lm cd sr lux cd m^2 m−^4 lx lm m−^2 becquerel s−^1 Bq s−^1 gray m^2 s−^2 Gy J kg−^1 sievertd^ m^2 s−^2 Sv J kg−^1 katale^ s−^1 mol kat s−^1 mol aThe radian and steradian may be used advantageously in expressions for derived units to distinguish between quantities of a different nature but of the same dimension; some examples are given in table 4. bIn practice, the symbols rad and sr are used where appropriate, but the derived unit ‘1’ is generally omitted. cIn photometry, the unit name steradian and the unit symbol sr are usually retained in expressions for derived units. dOther quantities expressed in sieverts are ambient dose equivalent, directional dose equiva- lent, personal dose equivalent, and organ equivalent dose. eThe 21st Conf´erence G´en´erale des Poids et Mesures decides to adopt the special name katal, symbol kat, for the SI unit mole per second to express catalytic activity, especially in the fields of medicine and biochemistry, ... (21th CGPM (1999), Resolution 12).
The SI derived unit of Ek is then m^2 kg s−^2 , which is given the special name “joule” and special symbol J.
2.3.1 Units for dimensionless quantities, quantities of dimension one
A derived quantity of dimension one, which is sometimes called a “dimensionless quantity”, is one for which all of the dimensional exponents are zero: dim Q = 1. It therefore follows that the derived unit for such a quantity is also the number one, symbol 1, which is sometimes called a “dimensionless derived unit”. Thus the SI unit of all quantities having the dimensional product one is the number one. Examples of such quantities are refractive index, relative permeability, and friction factor. All of these quantities are described as being dimensionless, or of dimension one, and have the coherent SI unit 1. Their values are simply expressed as numbers and, in general, the unit 1 is not explicitly shown.
For example: The mass fraction wB of a substance B in a mixture is given by wB = mB/m, where wB is the mass of B and m is the mass of the mixture. The dimension of wB is dim wB = M^1 M−^1 = 1; all of the dimensional exponents of wB are zero, and its derived unit is kg^1 kg−^1 = 1 also.
In a few cases, however, a special name is given to this unit, mainly to avoid confusion between some compound derived units. This is the case for the radian, steradian and neper.
2.4 Rules and style conventions for writing and using SI
unit symbols
The general principles concerning writing the unit symbols were adopted by the 9th CPGM (1948), by its Resolution 7:
- Roman (upright) type, in general lower case^4 , is used for the unit symbols. If, however, the name of the unit is derived from a proper name, the first letter of the symbol is in upper case.
- Unit symbols are unaltered in the plural.
- Unit symbols are not followed by a period^5.
To ensure uniformity in the use of the SI unit symbols, ISO International Standards give certain recommendations. Following these recommendations:
a) The product of two or more units are indicated by means of either a half-high (that is, centred) dot or a space^6. The half-high dot is preferred, because it is less likely to lead to confusion,
for example: N · m or N m. (^4) The recommended symbol for the litre (‘liter’) in the United States is L. (^5) Unless at the end of a sentence. (^6) ISO suggests that if a space is used to indicate units formed by multiplication, the space may be omitted if it does not cause confusion. This possibility is reflected in the common practice of using the symbol kWh rather than kW · h or kW h for the kilowatt hour.
b) A solidus (oblique stroke,/), a horizontal line, or negative exponents may be used to express a derived unit formed from two others by division,
for example: m/s, ms , or m s−^1
c) The solidus must not be repeated on the same line unless ambiguity is avoi- ded by parentheses. In complicated cases negative exponents or parentheses should be used,
for example: m/s^2 or m s−^2 but not: m/s/s m kg/(s^3 A) or m kg s−^3 A−^1 but not: m kg/s^3 /A
2.4.1 Space between numerical value and unit symbol
In the expression for the value of a quantity, the unit symbol is placed after the numerical value and a space is left between the numerical value and the unit symbol. The only exceptions to this rule are for the unit symbols for degree, minute, and second for plane angle: ◦, ′, and ′′, respectively (see Table 8), in which case no space is left between the numerical value and the unit symbol.
for example:
α = 30◦ 22 ′ 8 ′′^ Note: α is a quantity symbol for plane angle.
This rule means that the symbol ◦C for the degree Celsius is preceded by a space when one expresses the values of Celsius temperatures.
for example:
t = 30. 2 ◦C but not t = 30. 2 ◦C
3 SI Prefixes
3.1 Decimal multiples and submultiples of SI units
The 11th CGPM (1960), by its Resolution 12, adopted a first series of prefixes and symbols of prefixes to form the names and symbols of the decimal multiples and submultiples of SI units. Prefixes for 10−^15 and 10−^18 were added by the 12th CGPM (1964), by its Resolution 8, those for 10^15 and 10^18 by the CGPM (1975), by its Resolution 10, and those for 10^21 , 10^24 , 10−^21 , and 10−^24 were proposed by the CIPM for approval by the 19th CGPM (1991), and adopted. The prefixes are as shown in tabel 5.
3.2 Rules for using SI prefixes
In accord with the general principles adopted by the ISO^7 , the CIPM recommends that the following rules for using the SI prefixes be observed:
(^7) ISO 31, in ‘Units of measurement,’ ISO Standards Handbook 2, 2nd Edition, ISO, Geneva, 1982, pp. 17–
Table 6: — Prefixes for binary multiples — Factor Name Symbol Origin Derivation 210 kibi Ki kilobinary: (2^10 ) 1 kilo: (10^3 ) 1
220 mebi Mi megabinary: (2^10 ) 2 mega: (10^3 ) 2
230 gibi Gi gigabinary: (2^10 )^3 giga: (10^3 ) 3
240 tebi Ti terabinary: (2^10 )^4 tera: (10^3 ) 4
250 pebi Pi petabinary: (2^10 )^5 peta: (10^3 ) 5
260 exbi Ei exabinary: (2^10 )^6 exa: (10^3 ) 6
Table 7: — Examples and comparisons with SI prefixes — one kibibit 1 Kibit = 210 bit = 1 024 bit one kilobit 1 kbit = 103 bit = 1 000 bit one mebibyte 1 MiB = 220 B = 1 048 576 B one megabyte 1 MB = 106 B = 1 000 000 B one gibibyte 1 GiB = 230 B = 1 073 741 824 B one gigabyte 1 GB = 109 B = 1 000 000 000 B
3.2.2 The ‘degree Celsius’
Except for the kilogram, any SI prefix may be used with any SI unit, including the ‘degree Celsius’ and its symbol ◦C, for example: 10 −^3 ◦C = 1 m◦C (one millidegree Celsius), or 106 ◦C = 1 M◦C.
4 Prefixes for binary multiples
In December 1998 the International Electrotechnical Commission (IEC), the lea- ding international organization for worldwide standardization in electrotechnology, approved as an IEC International Standard names and symbols for prefixes for bi- nary multiples for use in the fields of data processing and data transmission. The prefixes are as shown in table 6. It is suggested that in English, the first syllable of the name of the binary-multiple prefix should be pronounced in the same way as the first syllable of the name of the corresponding SI prefix, and that the second syllable should be pronounced as “bee”.
Note
It is important to recognize that the new prefixes for binary multiples are not part of the International System of Units (SI), the modern metric system. However, for ease of understanding and recall, they were derived from the SI prefixes for positive powers of ten. As can be seen from the above table, the name of each new prefix is derived from the name of the corresponding SI prefix by retaining the first two letters of the name of the SI prefix and adding the letters “bi”, which recalls the word “binary”. Similarly, the symbol of each new prefix is derived from the symbol of the corresponding SI prefix by adding the letter “i”, which again recalls the word “binary”. (For consistency with the other prefixes for binary multiples, the symbol Ki is used for 2^10 rather than ki.)
4.1 Official publication
These prefixes for binary multiples, which were developed by IEC Technical Com- mittee (TC) 25, Quantities and units, and their letter symbols, with the strong support of the International Committee for Weights and Measures (CIPM) and the Institute of Electrical and Electronics Engineers (IEEE), were adopted by the IEC as Amendment 2 to IEC International Standard IEC 60027-2: Letter symbols to be used in electrical technology - Part 2: Telecommunications and electronics. The full content of Amendment 2, which has a publication date of 1999-01, is reflected in the tables above and the suggestion regarding pronunciation.
4.2 The binary.sty style for binary prefixes and (non-SI)
units
The binary.sty style for binary prefixes and (non-SI) units can be loaded by using the option binary, as in \usepackage[binary]{SIunits}. This unit should always be used in conjunction with the SIunits package.
5 Units outside the SI
Units that are outside the SI may be divided into three categories:
- those units that are accepted for use with the SI;
- those units that are temporarily accepted for use with the SI; and
- those units that are not accepted for use with the SI and thus must strictly be avoided.
5.1 Units accepted for use with the SI
The CIPM (1969) recognised that users of SI will also wish to employ with it certain units not part of it, but which are important and are widely used. These units are given in table 8. The combination of units of this table with SI units to form compound units should be restricted to special cases in order not to lose the advantage of the coherence of SI units. It is likewise necessary to recognise, outside the International System, some other units that are useful in specialised fields, because their values expressed in SI units must be obtained by experiment, and are therefore not known exactly (table 9).
5.2 Units temporarily accepted for use with the SI
Because of existing practice in certain fields or countries, in 1978 the CIPM con- sidered that it was permissible for the units given in table 10 to continue to be used with the SI until the CIPM considers that their use is no longer necessary. However, these units must not be introduced where they are not presently used.
6 Last notes about correct usage of the SI
The following points underline some of the important aspects about using SI units and their symbols, and also mention some of the common errors that are made.
